Hodgkin and Huxley

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Hodgkin and Huxley

Taken from: http://icwww.epfl.ch/~gerstner/SPNM/node14.html

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Hodgkin Huxley Model:

)()( tItIdtdVC inj

kk

m

)()()( tItItIk

kCinj withuQC and

dtdVC

dtduCIC

charging current

Ionchannels

)( xmxx VVgI

PkI k=gNa(Vmà VNa) +gK (Vmà VK ) +gL(Vmà VL)

C dtdVm= à gNa(Vmà VNa) à gK (Vmà VK ) à gL(Vmà VL) + I inj

General Membrane Equation (a very important Equation, used everywhere!)

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)()()( 43LmLKmKNamNa

kk VVgVVngVVhmgI

injLmLKmKNamNam IVVgVVngVVhmgdtdVC )()()( 43

PkI k=gNaf 1(t)(Vmà VNa) +gK f 2(t)(Vmà VK )+gLf 3(t)(Vmà VL)

Introducing time-dependence so as to get an Action Potential modelled

Following Hodgkin and Huxley (using rising AND falling functions):

Resulting time-dependent Membrane Equation

Hodgkin-Huxley Model: Action Potential / Threshold

Short, weak current pulses depolarize the cell only a little.

An action potential is elicited when crossing the threshold.

0 5 10 15 20t ms

806040200

2040

VV

m

Iinj = 0.42 nA

0 5 10 15 20t ms

806040200

2040

VV

m

Iinj = 0.43 nA

0 5 10 15 20t ms

806040200

2040

VV

m

Iinj = 0.44 nA

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Action Potential

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Action Potential

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huhuh

nununmumum

hh

nn

mm

)()1)((

)()1)(()()1)((

(for the giant squid axon)

)]([)(

10 uxx

ux

x

1

0

)]()([)(

)]()([)(

uuu

uuux

xxx

xx

x

with

• voltage dependent gating variables

time constant

asymptotic value

(u)

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)]([)(

10 uxx

ux

x

Solution:

x = exp(à üt) +x0

xç= à ü1exp(à ü

t)

Derivative

= à ü1 exp(à ü

t) +x0à x0

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• If u increases, m increases -> Na+ ions flow into the cell• at high u, Na+ conductance shuts off because of h• h reacts slower than m to the voltage increase• K+ conductance, determined by n, slowly increases with increased u

)]([)(

10 uxx

ux

x

action potential

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Let’s see it in action!

HHsim (seminar thema!)

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Your neurons surely don‘t like this guy!

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Voltage clamp method

• developed 1949 by Kenneth Cole• used in the 1950s by Alan Hodgkin and Andrew Huxley to measure

ion current while maintaining specific membrane potentials

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Voltage clamp method

Small depolarization

Ic: capacity currentIl: leakage current

Large depolarization

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The sodium channel (patch clamp)

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The sodium channel

Hodgkin-Huxley Model: Firing Latency

A higher current reduces the time until an action potential is elicited.

0 5 10 15 20t ms

806040200

2040

VV

m

Iinj = 0.45 nA

0 5 10 15 20t ms

806040200

2040

VV

m

Iinj = 0.65 nA

0 5 10 15 20t ms

806040200

2040

VV

m

Iinj = 0.85 nA

Hodgkin-Huxley Model: Firing Latency

A higher current reduces the time until an action potential is elicited.

0 5 10 15 20t ms

806040200

2040

VV

m

Iinj = 0.45 nA

0 5 10 15 20t ms

806040200

2040

VV

m

Iinj = 0.65 nA

0 5 10 15 20t ms

806040200

2040

VV

m

Iinj = 0.85 nA

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Function of the sodium channel

Hodgkin-Huxley Model: Refractory Period

Longer current pulses will lead to more action potentials.

However, the next action potential can only occur after a “waiting period” during which the cell return to its normal state.

This “waiting period” is called the refractory period.

0 5 10 15 20 25 30t ms

806040200

2040

VV

m

Iinj = 0.5 nA

0 5 10 15 20 25 30t ms

806040200

2040

VV

m

Iinj = 0.5 nA

0 5 10 15 20 25 30t ms

806040200

2040

VV

m

Iinj = 0.5 nA

Hodgkin-Huxley Model: Firing Rate

When injecting current for longer durations an increase in current strength will lead to an increase of the number of action potentials per time.

Thus, the firing rate of the neuron increases.

The maximum firing rate is limited by the absolute refractory period.

0 20 40 60 80 100t ms

806040200

2040

VV

m

Iinj = 0.2 nA

0 20 40 60 80 100t ms

806040200

2040

VV

m

Iinj = 0.3 nA

0 20 40 60 80 100t ms

806040200

2040

VV

m

Iinj = 0.6 nA

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Varying firing properties

???

Influence of steady hyperpolarization

Rhythmic burst in the absence of synaptic inputs

Influence of the neurotransmitter Acetylcholin

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Action Potential / Shapes:

Squid Giant Axon Rat - Muscle Cat - Heart

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Propagation of an Action Potential:

Action potentials propagate without being diminished (active process).

Distance

Time

Local current loops

Open channels per

mm2 m

embrane area


All sites along a nerve fiber will be depolarized until the potential passes threshold. As soon as this happens a new AP will be elicited at some distance to the old one.


All sites along a nerve fiber will be depolarized until the potential passes threshold. As soon as this happens a new AP will be elicited at some distance to the old one.

Main current flow is across the fiber.

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At the dendrite the incomingsignals arrive (incoming currents)

Molekules

Synapses

Neurons

Local Nets

Areas

Systems

CNS

At the soma currentare finally integrated.

At the axon hillock action potentialare generated if the potential crosses the membrane threshold

The axon transmits (transports) theaction potential to distant sites

At the synapses are the outgoing signals transmitted onto the dendrites of the target neurons

Structure of a Neuron:

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Chemical synapse

NeurotransmitterReceptors

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Neurotransmitters

Chemicals (amino acids, peptides, monoamines) that transmit, amplify and modulate signals between neuron and another cell.

Cause either excitatory or inhibitory PSPs.

Glutamate – excitatory transmitter

GABA, glycine – inhibitory transmitter

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Synaptic Transmission:

Synapses are used to transmit signals from the axon of a source to the dendrite of a target neuron.Synapses are used to transmit signals from the axon of a source to the dendrite of a target neuron.

There are electrical (rare) and chemical synapses (very common)

Synapses are used to transmit signals from the axon of a source to the dendrite of a target neuron.


At an electrical synapse we have direct electrical coupling (e.g., heart muscle cells).




At a chemical synapse a chemical substance (transmitter) is used to transport the signal.





Electrical synapses operate bi-directional and are extremely fast, chem. syn. operate uni-directional and are slower.






Chemical synapses can be excitatory or inhibitorythey can enhance or reduce the signalchange their synaptic strength (this is what happens during learning).

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Structure of a Chemical Synapse:

Motor Endplate (Frog muscle)

Axon

Synaptic cleft

Activezone

vesicles

Muscle fiber

Presynapticmembrane

Postsynapticmembrane

Synaptic cleft

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What happens at a chemical synapse during signal transmission:

The pre-synaptic action potential depolarises the axon terminals and Ca2+-channels open.Pre-synaptic

action potential

Concentration oftransmitterin the synaptic cleft

Post-synapticaction potential

The pre-synaptic action potential depolarises the axon terminals and Ca2+-channels open.

Ca2+ enters the pre-synaptic cell by which the transmitter vesicles are forced to open and release the transmitter.



Thereby the concentration of transmitter increases in the synaptic cleft and transmitter diffuses to the postsynaptic membrane.



Thereby the concentration of transmitter increases in the synaptic cleft and transmitter diffuses to the postsynaptic membrane.

Transmitter sensitive channels at the postsyaptic membrane open. Na+ and Ca2+ enter, K+ leaves the cell. An excitatory postsynaptic current (EPSC) is thereby generated which leads to an excitatory postsynaptic potential (EPSP).

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Neurotransmitters and their (main) Actions:

Transmitter Channel-typ Ion-current ActionTransmitter Channel-typ Ion-current Action

Acetylecholin nicotin. Receptor Na+ and K+ excitatory

Transmitter Channel-typ Ion-current Action


Glutamate AMPA / Kainate Na+ and K+ excitatory




GABA GABAA-Receptor Cl- inhibitory





Glycine Cl- inhibitory






Acetylecholin muscarin. Rec. - metabotropic, Ca2+ Release






Acetylecholin muscarin. Rec. - metabotropic, Ca2+ Release

Glutamate NMDA Na+, K+, Ca2+ voltage dependentblocked at resting potential

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Synaptic Plasticity

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At the dendrite the incomingsignals arrive (incoming currents)

Molekules

Synapses

Neurons

Local Nets

Areas

Systems

CNS

At the soma currentare finally integrated.

At the axon hillock action potentialare generated if the potential crosses the membrane threshold

The axon transmits (transports) theaction potential to distant sites

At the synapses are the outgoing signals transmitted onto the dendrites of the target neurons

Structure of a Neuron:

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Chemical synapse

NeurotransmitterReceptors

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Neurotransmitters

Chemicals (amino acids, peptides, monoamines) that transmit, amplify and modulate signals between neuron and another cell.

Cause either excitatory or inhibitory PSPs.

Glutamate – excitatory transmitter

GABA, glycine – inhibitory transmitter

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Synaptic Transmission:

Synapses are used to transmit signals from the axon of a source to the dendrite of a target neuron.Synapses are used to transmit signals from the axon of a source to the dendrite of a target neuron.



















Chemical synapses can be excitatory or inhibitorythey can enhance or reduce the signalchange their synaptic strength (this is what happens during learning).

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Sim ple Computational Operations that can be Performed with Neurons

A xon = O utpu t

Input 1

Input 2S om a =C P U

The system to be considered first:One Neuron receiving2 Synapses.

What are the computations that can be performed with such a simple system ?

Firs t th ings firs t: Basic OperationsA rithm etica l: + S um m ation

- S ubtraction. M ultip lication/ D iv is ion

Lociga l AN D O R N O T, e tc.

More Com pex Operations

C alcu lus: In tegra tion

dx/d t D iffe rentiation

L inear A lgebra :Vector O perationsy=A x M atrix O perations

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B e lieve it o r not: With a single neuron and 2 input you can compute all alrithmetic, many logic and some of the more complex operations !

R equ ire d R equ is its : 1) R esting P oten tia l (ca. -70 m v, constan t)2 ) F iring T hreshold3 ) E quilibrium P oten tia l o f d ifferen t ions4 ) Tim e-constants o f the ion-channels .

Summation

Keine Kognition ohne AdditionTransm itte r re lease a t a synapse leads to an excita tory postsynapticpotentia l (EP S P) because ion channe ls are open ing .

EPS P

m V

t

rest.pot.

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Necessary conditions for optim al sum m ation:1) synapses have to be close ly ad jacent2) p re-synaptic s igna ls have to arrive sim ultaneously3) resting potentia l and reversa l potentia l(s) have to be very d iffe rent.

E P S P = E PSP + E P S Pr e s A Bm V

t

re st.pot.

ABA

BThe little “shoulder” show s tha t theE PS P s were not true ly s im ultaneous.

Spatial Sum mation

E P S P < E P S P + E P S Pr e s A B

m V

t

rest.pot.

A B

A

B

Som a

D e ndrite

If the synapses are far from each o ther the am plitude w ill beless at the firs t sum m ing point. It w ill then further decayuntil reaching the som a.

Consider 1:

sim ultaneousinputs !

Sum m ationpoint

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H ow w ill the signa llook like at the sum m ation point ?

ABm V

t

rest.po t.

B S om aD endrite

A more complicated situationA1) The s ignal from B arrives la ter

a t the sum m ation po int because B is farther from it than A .2) The s ignal from B is sm aller a t the sum m ation poin t (sam e reason).

A

BS om a

Direction of signal propagation

The signal propagates essentia llyin a ll d irections. The directiontow ards the som a is (usua lly) theone wh ich is functiona lly re levant.

incomplete spatial summ ationEPSP = a EPSP + b EPSP ; b<a<1.0A Bres

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A

B

Consider 2: If the signals a re not s im ultaneous then the sum w ill be sm aller

m V

t

rest.po t.

A B

The early s ignal (A ) facilita tes the la ter signal (B ). Together the firing th resho ldm ight be reached but not a lone.

Temporal Summ ation

If the d ifference in arriva l tim es is too large, tem pora l sum m ation does no t occur anym ore !

m V

t

rest.po t.

A B

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A

B

Consider 3: If the equilibrium potentia l o f the invo lved ions is close to the resting potentia l then on ly incom ple te sum m ation is observed. Even a p la teau is possib le.

m V

t

rest.po t.

A B

The po tentia l o f the invo lved ions can never exceed the ir ow n equ ilibrium potentia l. (“C lipp ing”).

Conclusion: Summ ing with neurons is a rather com plex process. Spatial and temporal phenom ena and the potential levels will influence the result of the “sum mation” substantially.

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The sam e cond itions apply as for sum m ation. Then one can regardan IPSP as a s ign-inverted E PS P. “Sum m ation” becom es “Subtraction”.

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Special case: “shunting inhibition”The equilib rium poten tia l o f the ions “B ” is very c lose (”indentica l”)to the resting potentia l ! (A is exc itato ry as usual.)

E P S Pm V m V

t t

res t.po t.

res t.po t.

A B

H ow does the m em branepotentia l change ?

C l-

C l-open channel

Th is case is com m only observed for theC hloride ion.

W hat is the functional s ign ificance of th isbehavior ?

(almost)no potential change

W hen the a re opening (a lm ost) no ion current is obsered and thus the potentia l s tays (a lm ost) the sam e.

purp le channe ls

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The EPSP trave ls to the som a. The m em brane potentia l w ill be depolarized a long the w ay.

W hat happens at location C l w ith the re la tion betweenm em brane po tentia l and C l-equilibrium potentia l ?

A C l-current is the consequence. The positive m em brane pot. fluctua tion (viz. EPSP ) w ill be im m ediate ly com pensated for. Thus, a t the open C l channe ls no m ore depo larization is observed . The E PS P is e lectrically shunted !

Functional significance of “shunting inhibition”C onsider the case w ere C l-channe ls a re a lready open w hen theexc ita tory channels A are opening and an EPSP is e lic ited there.

A

Clto thesom a

to the periphera ldendrite

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The physio log ica l transm itte r is G lu tam ate (G lu ).

i n

o u t

i n

o u t

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Hodgkin and Huxley